Learning in communication systems

ABSTRACT

An apparatus, method and computer program is described comprising: initialising trainable parameters of a transmission system having a transmitter, a channel and a receiver; generating training symbols on the basis of a differentiable distribution function; transmitting modulated training symbols to the receiver over the channel in a training mode; generating a loss function based on the generated training symbols and the modulated training symbols as received at the receiver of the transmission system; and generating updated parameters of the transmission system in order to minimise the loss function.

FIELD

The present specification relates to learning in communication systems.

BACKGROUND

A simple communications system includes a transmitter, a transmissionchannel, and a receiver. The choice of communication scheme used in thetransmission of data from the transmitter to the receiver can have asignificant impact of the end-to-end performance of such communicationssystems. Although developments have been made, there remains scope forfurther developments in this area.

SUMMARY

In a first aspect, this specification describes an apparatus comprising:means for initialising (e.g. randomly, or in some other way—such as onthe basis of channel information) trainable parameters of a transmissionsystem, wherein the transmission system comprises a transmitter, achannel and a receiver, wherein the transmitter includes a transmitteralgorithm (e.g. implemented as a neural network) for converting one ormore inputs (e.g. data bit(s) or a data stream) into one or more datasymbols and a modulator for converting said data symbols into transmitsymbols in accordance with a modulation scheme, wherein the trainableparameters of the transmission system include a probability function(e.g. implementing probabilistic shaping, as described herein) defininga probability that individual data symbols are output by the transmitteralgorithm; means for generating training symbols on the basis of adistribution function, wherein the distribution function isdifferentiable; means for transmitting modulated training symbols to thereceiver over the channel in a training mode; means for generating aloss function based on the generated training symbols and the modulatedtraining symbols as received at the receiver of the transmission system;means for generating updated parameters of the transmission system (e.g.using machine learning principles, for example by updating a neuralnetwork) in order to minimise the loss function, wherein generatingupdated parameters of the transmission system comprise updating theparameters of the transmitter algorithm; and means for repeating thegenerating the training symbols, generating the loss function andupdating parameters of the transmission system until a first conditionis reached. The data symbols may be a finite and discrete set ofsymbols. The data symbols may correspond to constellation points of thetransmitter. The probability function may be a discrete function that isnot differentiable (and may be replaced by a differentiableapproximation (the distribution function) during training). Thegeneration of training symbols on the basis of a differentiabledistribution function may enable the use of back-propagation during thetraining of parameters.

Updating the parameters of the training algorithm may include trainingthe probability function. Updating said parameters may also includeupdating other parameters, such as the modulator constellation.

The distribution function may be an approximation of the probabilityfunction. For example, the distribution function may be a differentiableapproximation of the probability function (which probability functionmay be discrete and non-differentiable).

The training symbols may be generated by a Gumbel-softmax algorithm,such as using Gumbel-softmax reparameterization.

The transmit symbols (and the training symbols in the training mode) maycorrespond to fixed constellation positions of a modulation schemeimplemented by said modulator. Alternatively, the transmit symbols (andthe training symbols in the training mode) may correspond to variableconstellation positions of a modulation scheme implemented by saidmodulator, wherein the trainable parameters of the transmission systemcomprise constellation point positions of said modulation scheme.

The receiver may be configured to receive said transmit symbols astransmitted over said channel in an operational mode and to receive saidmodulated training symbols as transmitted over the channel in a trainingmode.

Generating updated parameters of the transmission system may compriseupdating parameters of a trainable receiver algorithm. Thus, forexample, end-to-end training of the transmission system may be provided.

Some embodiments may include means for updating the trainable parametersof the transmission system using the generated updated parameters of thetransmission system. For example, the transmitter parameters may beupdated in a training mode and then deployed to the transmitteralgorithm. The trainable parameters of the transmitter system maycomprise one of more of: trainable parameters of the transmitteralgorithm; trainable parameters of the modulator; and trainableparameters of the receiver.

The probability function and the distribution function may be at leastpartially dependent on channel information (such as signal-to-noiseratio) of the transmission system.

In some embodiments, the first condition comprises a defined performancelevel. Alternatively, or in addition, the first condition may comprise adefined number of iterations.

The transmitter algorithm may be implemented as a look-up table in anoperational mode.

The loss function may be related to one or more of block error rate, biterror rate, mutual information and categorical cross-entropy.

The means for generating updated parameters of the transmission systemmay optimise one or more of a batch size of symbol approximations usedin the training mode and a learning rate.

The means for generating updated parameters of the transmission systemmay update said parameters using stochastic gradient descent (or someother backpropagation algorithm).

The transmitter may comprise a transmitter neural network configured toimplement said transmitter algorithm. A receiver algorithm may include areceiver neural network.

The said means may comprise: at least one processor; and at least onememory including computer program code, the at least one memory and thecomputer program configured, with the at least one processor, to causethe performance of the apparatus.

In a second aspect, this specification describes a method comprising:initialising trainable parameters of a transmission system, wherein thetransmission system comprises a transmitter, a channel and a receiver,wherein the transmitter includes a transmitter algorithm for convertingone or more inputs into one or more data symbols and a modulator forconverting said data symbols into transmit symbols in accordance with amodulation scheme, wherein the trainable parameters of the transmissionsystem include a probability function defining a probability thatindividual data symbols are output by the transmitter algorithm;generating training symbols on the basis of a distribution function,wherein the distribution function is differentiable; transmittingmodulated training symbols to the receiver over the channel in atraining mode; generating a loss function based on the generatedtraining symbols and the modulated training symbols as received at thereceiver of the transmission system; generating updated parameters ofthe transmission system in order to minimise the loss function, whereingenerating updated parameters of the transmission system compriseupdating the parameters of the transmitter algorithm; and repeating thegenerating the training symbols, generating the loss function andupdating parameters of the transmission system until a first conditionis reached.

The training symbols may be generated by a Gumbel-softmax algorithm.

The transmit symbols may correspond to fixed constellation positions ofa modulation scheme implemented by said modulator. Alternatively, thetransmit symbols may correspond to variable constellation positions of amodulation scheme implemented by said modulator, wherein the trainableparameters of the transmission system comprise constellation pointpositions of said modulation scheme.

Generating the updated parameters of the transmission system maycomprise updating parameters of a trainable receiver algorithm.

Some embodiments include updating the trainable parameters of thetransmission system using the generated updated parameters of thetransmission system. The said trainable parameters of the transmittersystem may comprise one of more of: trainable parameters of thetransmitter algorithm; trainable parameters of the modulator; andtrainable parameters of the receiver.

The probability function and the distribution function may be at leastpartially dependent on channel information of the transmission system.

The second aspect may further comprise at least some of the features ofthe first aspect as described above.

In a third aspect, this specification describes an apparatus configuredto perform any method as described with reference to the second aspect.

In a fourth aspect, this specification describes a work productcomprising a look up table or array, created by means of the method ofthe second aspect.

In a fifth aspect, this specification describes computer-readableinstructions which, when executed by computing apparatus, cause thecomputing apparatus to perform any method as described with reference tothe third or fourth aspects.

In a sixth aspect, this specification describes a computer readablemedium comprising program instructions stored thereon for performing atleast the following: initialising trainable parameters of a transmissionsystem, wherein the transmission system comprises a transmitter, achannel and a receiver, wherein the transmitter includes a transmitteralgorithm for converting one or more inputs into one or more datasymbols and a modulator for converting said data symbols into transmitsymbols in accordance with a modulation scheme, wherein the trainableparameters of the transmission system include a probability functiondefining a probability that individual data symbols are output by thetransmitter algorithm; generating training symbols on the basis of adistribution function, wherein the distribution function isdifferentiable; transmitting modulated symbol to the receiver over thechannel in a training mode; generating a loss function based on thegenerated symbol and the modulated symbol as received at the receiver ofthe transmission system; generating updated parameters of thetransmission system in order to minimise the loss function, whereingenerating updated parameters of the transmission system compriseupdating the parameters of the transmitter algorithm; and repeating thegenerating the symbol approximations, generating the loss function andupdating parameters of the transmission system until a first conditionis reached.

In an seventh aspect, this specification describes a computer programcomprising instructions for causing an apparatus to perform at least thefollowing: initialise trainable parameters of a transmission system,wherein the transmission system comprises a transmitter, a channel and areceiver, wherein the transmitter includes a transmitter algorithm forconverting one or more inputs into one or more data symbols and amodulator for converting said data symbols into transmit symbols inaccordance with a modulation scheme, wherein the trainable parameters ofthe transmission system include a probability function defining aprobability that individual data symbols are output by the transmitteralgorithm; generate training symbols on the basis of a distributionfunction, wherein the distribution function is differentiable; transmitmodulated symbol to the receiver over the channel in a training mode;generate a loss function based on the generated symbol and the modulatedsymbol as received at the receiver of the transmission system; generateupdated parameters of the transmission system in order to minimise theloss function, wherein generating updated parameters of the transmissionsystem comprise updating the parameters of the transmitter algorithm;and repeat the generating the symbol approximations, generating the lossfunction and updating parameters of the transmission system until afirst condition is reached.

In an eighth aspect, this specification describes an apparatuscomprising: at least one processor; and at least one memory includingcomputer program code which, when executed by the at least oneprocessor, causes the apparatus to: initialise trainable parameters of atransmission system, wherein the transmission system comprises atransmitter, a channel and a receiver, wherein the transmitter includesa transmitter algorithm for converting one or more inputs into one ormore data symbols and a modulator for converting said data symbols intotransmit symbols in accordance with a modulation scheme, wherein thetrainable parameters of the transmission system include a probabilityfunction defining a probability that individual data symbols are outputby the transmitter algorithm; generate training symbols on the basis ofa distribution function, wherein the distribution function isdifferentiable; transmit modulated symbol to the receiver over thechannel in a training mode; generate a loss function based on thegenerated symbol and the modulated symbol as received at the receiver ofthe transmission system; generate updated parameters of the transmissionsystem in order to minimise the loss function, wherein generatingupdated parameters of the transmission system comprise updating theparameters of the transmitter algorithm; and repeat the generating thesymbol approximations, generating the loss function and updatingparameters of the transmission system until a first condition isreached.

In a ninth aspect, this specification describes an apparatus comprising:a initialisation module for initialising (e.g. randomly, or in someother way—such as on the basis of channel information) trainableparameters of a transmission system, wherein the transmission systemcomprises a transmitter, a channel and a receiver, wherein thetransmitter includes a transmitter algorithm (e.g. implemented as aneural network) for converting one or more inputs (e.g. data bit(s) or adata stream) into one or more data symbols and a modulator forconverting said data symbols into transmit symbols in accordance with amodulation scheme, wherein the trainable parameters of the transmissionsystem include a probability function (e.g. implementing probabilisticshaping, as described herein) defining a probability that individualdata symbols are output by the transmitter algorithm; a training module(such as a shaping device) for generating training symbols on the basisof a distribution function, wherein the distribution function isdifferentiable; a transmitter for transmitting modulated trainingsymbols to the receiver over the channel in a training mode; a processorfor generating a loss function based on the generated training symbolsand the modulated training symbols as received at the receiver of thetransmission system; an updating module for generating updatedparameters of the transmission system (e.g. using machine learningprinciples, for example by updating a neural network) in order tominimise the loss function, wherein generating updated parameters of thetransmission system comprise updating the parameters of the transmitteralgorithm; and a control module for repeating the generating thetraining symbols, generating the loss function and updating parametersof the transmission system until a first condition is reached. The datasymbols may be a finite and discrete set of symbols. The data symbolsmay correspond to constellation points of the transmitter. Theprobability function may be a discrete function that is notdifferentiable (and may be replaced by a differentiable approximation(the distribution function) during training).

BRIEF DESCRIPTION OF THE DRAWINGS

Example embodiments will now be described, by way of non-limitingexamples, with reference to the following schematic drawings, in which:

FIG. 1 is a block diagram of an example end-to-end communication systemin accordance with an example embodiment;

FIG. 2 is a block diagram of an example transmitter that may be used inthe communication system of FIG. 1 ;

FIG. 3 is a block diagram of an example receiver that may be used in thecommunication system of FIG. 1 ;

FIG. 4 is a block diagram of a transmission system in accordance with anexample embodiment;

FIG. 5 is a block diagram of a transmitter in accordance with an exampleembodiment;

FIG. 6 is a block diagram of an example receiver module in accordancewith an example embodiment;

FIG. 7 is a flow chart showing an algorithm in accordance with anexample embodiment;

FIG. 8 is a block diagram of a components of a system in accordance withan exemplary embodiment; and

FIGS. 9 a and 9 b show tangible media, respectively a removable memoryunit and a compact disc (CD) storing computer-readable code which whenrun by a computer perform operations according to embodiments.

DETAILED DESCRIPTION

The scope of protection sought for various embodiments of the inventionis set out by the independent claims. The embodiments and features, ifany, described in the specification that do not fall under the scope ofthe independent claims are to be interpreted as examples useful forunderstanding various embodiments of the invention.

In the description and drawings, like reference numerals refer to likeelements throughout.

FIG. 1 is a block diagram of an example end-to-end communication system,indicated generally by the reference numeral 1, in accordance with anexample embodiment. The system 1 includes a transmitter 2, a channel 4and a receiver 6. Viewed at a system level, the system 1 converts inputbits (b) received at the input to the transmitter 2 into output bits({circumflex over (b)}) at the output of the receiver 6. Morespecifically, the transmitter 2 converts the input bits (b) intotransmit symbols (x) for transmission over the channel 4 and thereceiver 6 generates the output bits ({circumflex over (b)}) fromsymbols (y) received from the channel 4.

FIG. 2 is a block diagram of an example transmitter 2 that may be usedin the communication system 1 described above. As shown in FIG. 2 , thetransmitter 2 includes a bit-to-symbols module 12 and a modulator 14.The bits-to-symbols module 12 receives the input data bits (b) andconverts those bits into data symbols (s) for transmission. For example,the input data bits (b) may take the form of a data stream, which datastream is packaged for transmission by the bits-to-symbol module 12. Themodulator 14 converts the data symbols into transmit symbols (x) inaccordance with a modulation scheme. The transmit symbols are thentransmitted over the channel 4 and received at the receiver 6 asreceived symbols (y).

FIG. 3 is a block diagram of an example receiver 6 that may be used inthe communication system 1 described above. As shown in FIG. 3 , thereceiver 6 includes a demodulator 22 and demapper module 24. Thedemodulator 22 converts the received symbols (y) into symbolprobabilities p_(Ø)(s|y) and the demapper 24 converts the symbolprobabilities into bit probabilities p(b_(i)|y).

A number of modulation techniques could be used in the implementation ofthe modulator 14 (and the demodulator 22). These include amplitude shiftkeying (ASK) in which the amplitude of a carrier signal is modifiedbased on a signal being transmitted and phase shift keying (PSK) inwhich the phase of a carrier signal is modified based on a signal beingtransmitted. By way of example, quadrature phase shift keying (QPSK) isa form of phase shift keying in which two bits are modulated at once,selecting one of four possible carrier phases shifts (e.g. 0, +90degree, 180 degrees, −90 degrees). Such carrier phase and amplitudes areoften represented as constellation positions in a complex plane. Theskilled person will be aware of many other suitable modulationtechniques.

The choice of modulation scheme used to transmit information such as thecommunication system 1 has an impact on the end-to-end performance ofsuch communication systems. Moreover, such modulation schemes may beoptimised.

For communication schemes having multiple constellation points (such asamplitude and/or phase shift keying modulation schemes), the locationsof such constellation points in a complex plane may be optimised.Alternatively, or in addition, the relative frequency with which suchconstellation points are used may be optimised. Those methods arereferred to herein as geometric shaping and probabilistic shapingrespectively.

FIG. 4 is a block diagram of a transmission system, indicated generallyby the reference numeral 30, in accordance with an example embodiment.The transmission system 30 comprises a transmitter 32, a channel 34 anda receiver 36 (that have some similarities with the transmitter 2,channel 4 and receiver 6 described above).

As shown in FIG. 4 , the transmitter 32 comprises a bits-to-symbolmodule 37 (similar to the bits-to-symbol module 12 described above), amodulator 38 (similar to the modulator 14) and a shaping device 39. Thereceiver 36 comprises a demodulator 40 (similar to the demodulator 22)and a demapper module 41 (similar to the demapper module 24).

As discussed further below, the transmitter 32 maps an input stream ofbits onto transmit symbols. The transmitter includes a neural network(or some other function with trainable parameters).

The bits-to-symbol module 37 receives an incoming bit sequence of sizeN, denoted by b=[b₁, . . . , b_(N)], and maps that bit sequence ontohypersymbols s∈

, such that the frequencies of occurrence of symbols s correspond to aprobability distribution p_(θ)(s) provided by the shaping device 39.Here,

={1, . . . , K} is the finite and discrete set of hypersymbols. Thehypersymbol s is provided to the modulator 38, which maps s onto complextransmit symbols x∈

^(M), where M is the number of channel uses.

The channel 34 of the system 30 takes the complex transmission symbolsx∈

^(M) and outputs M received samples pooled in y∈

^(M). The possibly unknown transition probability p(y|x) describes theinput-output relation of the channel 34.

The receiver 36 of the system 30 operates on the sequence y∈

^(M) of M samples generated by the channel 34. The demodulator 40 of thereceiver 36 maps the received symbols onto symbol probabilitiesp_(ϕ)(s|y), and the demapper module 41 outputs bit probabilitiesp(b_(i)|y).

FIG. 5 is a block diagram of a transmitter, indicated generally by thereference numeral 50, in accordance with an example embodiment. Thetransmitter 50 is an example of the transmitter 32 described above; inparticular, the transmitter 50 shows an example arrangement for trainingthe transmitter 32 described above.

The transmitter 50 comprises the bits-to-symbol module 37, the modulator38 and the shaping device 39 in accordance with an example embodiment.The shaping device 39 may be used during training of the transmitter 50.As described further below, in a normal operation mode, the shapingdevice 39 may be replaced, for example with a look-up-table providingthe appropriate settings.

The modulator 38 comprises an embedding module 52, a normalizationmodule 54 and real-to-complex conversion module 56. The shaping device39 comprises a parameters module 60, a summing module 62, a Gumbelsampler 63, a softmax module 64, a softmax τ module 65 and astraight-through estimator 66.

The bits-to-symbol module 37 maps the incoming bits b onto thehypersymbols s. This mapping is implemented such that the frequency withwhich the symbols are used matches a given probability distributionp_(θ)(s). The probability distribution p_(θ) (s) is generated by theshaping module 39, as discussed further below.

The bits-to-symbols module 37 may be implemented in a number of ways. Byway of example, a constant composition distribution matching (CCDM)approach may be used. CCDM involves mapping a vector of N₁ bits to avector of N₂ symbols, where N₁ and N₂ are typically large numbers. Fromthe set of all possible vectors of symbols of size N₂, one selects theones such that symbols appear approximately with the desired targetprobabilities. These selected vectors form the set

. The larger is N₂, the better is the approximation of the targetprobabilities. N₁ must be such that N₁≤log₂|

|. Then, one maps the vectors of bits of size N₁ to the vectors ofsymbols from g using arithmetic coding.

As discussed in detail below, it may be desired to train the system 50using machine-learning principles, such as backpropagation. Theoperation performed by the bits-to-symbol module 37 is a discrete andnon-differentiable function, and thus, many training techniques used inmachine learning are not applicable.

The shaping device 39 may be implemented as a neural network withparameters θ (or any other trainable function). As suggested in FIG. 5 ,the shaping device 39 may make use of a straight-through Gumbel-softmaxreparameterisation technique. The Gumbel-softmax technique states thatsamples s∈

of any discrete probability distribution p_(θ)(s) can be approximatelysampled by:

$\begin{matrix}{{Where}\mspace{275mu}{\overset{˜}{s} = {{soft}{\max\left( \frac{g + z}{\tau} \right)}}}} & (1) \\{\mspace{245mu}{{{softmax}(a)} = \left\lbrack \frac{\exp\; a_{i}}{\sum\limits_{j}{\exp a_{j}}} \right\rbrack_{i}}} & (2)\end{matrix}$and g_(i) with i∈

are samples from a Gumbel (0,1) distribution, and the {z_(i)} with i∈

are the so-called logits (e.g. the logarithm of the unnormalisedprobabilities—the softmax function reverses this process) computed by atrainable neural network with parameters θ. (The parameter T isdiscussed further below.)

The distribution p_(θ)(s) can be retrieved from the logits byp_(θ)=softmax(z). In the system 50, an example architecture of theneural network generating the logits is depicted. The logits aredirectly trained in the example, and therefore θ=z. Note that otherarchitectures are possible. For example, the logits could be computerfrom channel information (such as signal-to-noise ratio) by a neuralnetwork.

By optimizing θ, one optimizes the distribution p_(θ)(s).

If the argmax operator were used instead of the softmax operator in theequation (1) above, this would enable sampling from the exactdistribution p_(θ)(s). The argmax output would be converted to a one-hotrepresentation, i.e., a vector with a single element set to one and theothers set to zero. The index of the element set to one would correspondto the output of the argmax. However, the argmax operator is notdifferentiable, preventing the optimization of p_(θ)(s) by usualstochastic gradient descent (SGD) or similar backpropagation techniques.The softmax function may be used as a differentiable approximation ofthe argmax operator. The positive parameter T controls how largely thedistribution of the generated samples {tilde over (s)} deviates from theexact distribution p_(θ)(s). Setting τ to low values enables betterapproximation, but makes the training harder as it increases thesharpness of the softmax τ layer, making backpropagation of the gradientmore difficult. A drawback of the softmax function is that it outputsonly an approximation of one-hot vectors. To ensure that at training theinput of the modulator is always a true one-hot vector, the output ofthe softmax may be fed into a straight-through estimator 66. Astraight-through estimator is a very simple unit which discretizes itsinput to the closest truly one-hot vector (by setting its largestelement equal to one and all other elements to zero), but can be skippedwhen performing backpropagation (i.e., the discretization is ignored).

Stochastic gradient descent seeks to optimize the trainable parametersbased on backpropagating gradients. Since, in the arrangement describedabove, only the proposed sampling device but not the bits-to-symbolsmodule 37 is trainable, different submodules are active at training andat deployment. This is depicted by the switch 58 in the system 50.

During training, the trainable shaping device 39 is active, and thehypersymbols s are generated by this device. In order to learn adistribution p_(θ)(s) that maximizes the information rate, the flow ofbackpropagating gradients is fed into the trainable device and theparameters θ are updated accordingly. Once training is complete, thelearned distribution p_(θ)(s) is fed into the bits-to-symbols module 37and the switch 58 is set to “evaluation”. Thus, at deployment, thesymbols are now generated by the bits-to-symbols module 37, which mapsthe incoming bit stream onto hypersymbols with the learned frequencyp_(θ)(s). As noted above, such an arrangement may be implemented by alook-up-table or some similar module.

The modulator 38 maps the symbols s onto constellation points x in thecomplex plane. The actual location of these points can either be fixedor jointly optimized with p_(θ)(s). If the constellation is alsolearned, the modulator 38 can implemented as a neural network withtrainable parameters (or any other learnable function), and denoted byg_(ψ). An example architecture of a trainable modulator is shown in FIG.5 , which includes an embedding layer 52 of dimension K×2M and anormalization layer 54 that takes as a second input the learneddistribution p_(θ)(s) to ensure some power constraint, e.g., E{∥x∥₂²}=M. Finally, a real-to-complex conversion module 56 maps the 2M realoutputs to M complex channel symbols. The skilled person will be awareof alternative implementations of the modulator 38 (particularly if thelocations of the constellation points are fixed).

The systems 30 and 50 may be trained using end-to-end learning. Withthis approach, the transmitter 32, the channel 34 and the receiver 36may be implemented as a single neural network, and the end-t0-end systemtrained to reconstruct the input(s) and the output. In some embodiments,a differential channel model may be assumed to be available. Alternativeembodiments are possible, for example, in the event that no suitablechannel model is available.

With end-to-end training, a trainable receiver r_(Ø) with parameters Øimplemented as a neural network (or some other trainable function) maybe provided.

FIG. 6 is a block diagram of an example receiver module, indicatedgenerally by the reference numeral 70, in accordance with an exampleembodiment. The receiver module 70 comprises a complex-to-realconversion module 71, a first dense layer 72, a second dense layer 73and a softmax layer 74. The provision of two dense layers in the system70 is an example only; any number of dense layers could be provided.

The complex-to-real conversion module 71 converts a received vector y∈

^(M) into real values. This can be done, for example, by concatenatingthe real and imaginary parts of a sample to obtain a vector that takesvalues in

^(2M).

The so-obtained vector is fed provided to the dense layers 72 and 73,which dense layers may have different activation functions (e.g. ReLU,tanh, sigmoid, linear etc.). The softmax layer last layer has |

| output dimensions and leverages softmax activation functions togenerate a discrete probability mass function p_(ϕ)(s|y), whose elementscan be interpreted as the probability that the symbol s was transmitted.

To determine one or more of the trainable function θ (related toprobabilistic shaping), φ (related to constellation positions orgeometric shaping) and Ø (related to the receiver), stochastic gradientdescent (or some similar algorithm) is performed. One possible lossfunction which could be minimised is the cross-entropy (CE) lossfunction to which is subtracted the symbols entropy:

=−E _(s˜p) _(θ) _((s),y˜p(y|g) _(ψ) _((s))){log p _(ϕ)(s|y)}+Σ_(s) p_(θ)(s)log p _(θ)(s)  (3)

In the equation (3), the expectation is taken over all realisations ofthe symbols s and channel outputs y. Assuming independent andidentically distributed (i.i.d.) realisations, the loss function can beestimated by:

$\begin{matrix}{{\mathcal{L}\left( {\theta,\varphi,\phi} \right)} \approx {{{- \frac{1}{B}}{\sum\limits_{i = 1}^{B}{\log{p_{\phi}\left( s^{(i)} \middle| y^{(i)} \right)}}}} + {\sum\limits_{s}{{p_{\theta}(s)}\log{p_{\theta}(s)}}}}} & (4)\end{matrix}$

Where B denotes the batch size (i.e. the number of training examplesused to approximate the loss). Note that optimisation would not beconducted over φ if the modulator is assumed to have fixedconstellations.

FIG. 7 is a flow chart showing an algorithm, indicated generally by thereference numeral 80, in accordance with an example embodiment.

The algorithm 80 starts at operation 81, wherein trainable parameters ofa transmission system (such as the transmission system discussed abovewith references to FIGS. 4 and 5 ) are initialised. The initialisationmay take many forms, from example, the initialisation may set thetrainable parameters randomly (or pseudo-randomly) or in some arbitraryway; alternatively, the trainable parameters may be set to a predefinedstarting point or defined according to some algorithm (e.g. on the basisof channel information of the transmission system, such assignal-to-noise ratio).

The initialisation of parameters may be implemented by the parametersmodule 60. The parameters module 60 may, for example, set an initialprobability function defining a probability that individual data symbols(e.g. constellation points of the transmitter) are output by thetransmitter algorithm. The probability function is a discrete functionand is not directly differentiable.

The algorithm moves to operation 82, where training symbols aregenerated on the basis of a distribution function, wherein thedistribution function is differentiable. The distribution function maybe an approximation (e.g. a differentiable approximation) of theprobability function. As noted above, the probability function istypically a discrete and non-differential probability function, which isnot conducive to backpropagation techniques that are often employed inmachine-learning processes (as discussed further below). Thus, anon-differentiable probability function may be replaced by adifferential approximation during training.

The training symbols may be generated by a Gumbel-softmax algorithm(such as using Gumbel-softmax reparameterization discussed above). Thetraining symbols, as generated by the shaping device 39 in the trainingmode, are provided to the modulator 38 and modulated training symbolsare transmitted to the receiver 36 over the channel 34.

As discussed above, the training symbols (in the training mode) maycorrespond to fixed constellation positions of a modulation schemeimplemented by the modulator 38. Similarly, the transmit symbols (in thenormal operation mode) may correspond to fixed constellation positions.

Alternatively, the training symbols, in the training mode (and thetransmit symbols in the normal operation mode) may correspond tovariable constellation positions of a modulation scheme implemented bysaid modulator 38. Moreover, the trainable parameters of thetransmission system may comprise constellation point positions of saidmodulation scheme.

The symbols that are generated and transmitted over said channel in theoperation 82 are received by the receiver 36.

At operation 83, a loss function is generated based on the generatedtraining symbols and the modulated training symbols as received at thereceiver of the transmission system. The loss function may take manydifferent forms. For example, the loss function may be related to one ormore of block error rate, bit error rate, mutual information andcategorical cross-entropy.

At operation 84, the parameters of the transmission system are updatedin order to minimise the loss function, wherein generating updatedparameters of the transmission system comprise updating the parametersof the transmitter algorithm (e.g. using machine-learning principles,for example by updating a neural network). Generating updated parametersof the transmission system may include training the probability functiondiscussed above. Thus, for example, the probability function may betrained in order to set the probabilistic shaping such that the biterror rate (or some other error function) is minimised. Other parametersmay also be updated, such as the modulator constellation positions.

In addition to updating the trainable transmitter algorithm, generatingupdated parameters of the transmission system in the operation 84 maycomprise updating parameters of a trainable receiver algorithm (therebyimplementing end-to-end training of the transmission system).

At operation 85, it is determined whether the algorithm 80 is complete.If so, the algorithm terminates at operation 86; otherwise, thealgorithm returns to operation 82 such that the operations 82 to 85 arerepeated (i.e. the generating and transmitting of training symbols, thegenerating a loss function and updating of parameters of thetransmission system are repeated).

The algorithm may be deemed to be complete (at operation 85) when acondition is reached. Such a condition may take many forms. For example,the operations 82 to 85 may be repeated until a defined performancelevel is reached. Alternatively, or in addition, the operations 82 to 85may be repeated until a defined number of iterations has been completed.

It should also be noted that the batch size B of symbol approximationsused in the training made and/or the learning rate (and possibly otherparameters of the chosen SGD variant, e,g, ADAM, RMSProp, Momentum)could be optimization parameters of the algorithm 80.

Once trained, the trainable parameters of the transmission system may bedeployed. This may involve updating the trainable parameters of thetransmission system using the generated updated parameters of thetransmission system. Moreover, the trainable parameters of thetransmitter system may comprise one of more of: trainable parameters ofthe transmitter algorithm, trainable parameters of the modulator andtrainable parameters of the receiver.

A number of modifications to the example embodiments described above arepossible. For example, end-to-end training of the transmission systems30 and 50 could be performed without knowledge of the relevant channelmode, and without requiring a differentiable channel mode, by leveragingprinciples of reinforcement learning. Moreover, a differential channelmodel could be learned using the principles of generative adversarialnetworks. Once such a model has been learned, end-to-end learning couldbe performed using the learned channel model as a differentiable channelmodel.

For completeness, FIG. 8 is a schematic diagram of components of one ormore of the modules described previously (e.g. the transmitter orreceiver neural networks), which hereafter are referred to genericallyas processing systems 110. A processing system 110 may have a processor112, a memory 114 closely coupled to the processor and comprised of aRAM 124 and ROM 122, and, optionally, hardware keys 120 and a display128. The processing system 110 may comprise one or more networkinterfaces 118 for connection to a network, e.g. a modem which may bewired or wireless.

The processor 112 is connected to each of the other components in orderto control operation thereof.

The memory 114 may comprise a non-volatile memory, a hard disk drive(HDD) or a solid state drive (SSD). The ROM 122 of the memory 114stores, amongst other things, an operating system 125 and may storesoftware applications 126. The RAM 124 of the memory 114 is used by theprocessor 112 for the temporary storage of data. The operating system125 may contain code which, when executed by the processor, implementsaspects of the algorithm 80.

The processor 112 may take any suitable form. For instance, it may be amicrocontroller, plural microcontrollers, a processor, or pluralprocessors.

The processing system 110 may be a standalone computer, a server, aconsole, or a network thereof.

In some embodiments, the processing system 110 may also be associatedwith external software applications. These may be applications stored ona remote server device and may run partly or exclusively on the remoteserver device. These applications may be termed cloud-hostedapplications. The processing system 110 may be in communication with theremote server device in order to utilize the software application storedthere.

FIGS. 9A and 9B show tangible media, respectively a removable memoryunit 165 and a compact disc (CD) 168, storing computer-readable codewhich when run by a computer may perform methods according toembodiments described above. The removable memory unit 165 may be amemory stick, e.g. a USB memory stick, having internal memory 166storing the computer-readable code. The memory 166 may be accessed by acomputer system via a connector 167. The CD 168 may be a CD-ROM or a DVDor similar. Other forms of tangible storage media may be used.

Embodiments of the present invention may be implemented in software,hardware, application logic or a combination of software, hardware andapplication logic. The software, application logic and/or hardware mayreside on memory, or any computer media. In an example embodiment, theapplication logic, software or an instruction set is maintained on anyone of various conventional computer-readable media. In the context ofthis document, a “memory” or “computer-readable medium” may be anynon-transitory media or means that can contain, store, communicate,propagate or transport the instructions for use by or in connection withan instruction execution system, apparatus, or device, such as acomputer.

Reference to, where relevant, “computer-readable storage medium”,“computer program product”, “tangibly embodied computer program” etc.,or a “processor” or “processing circuitry” etc. should be understood toencompass not only computers having differing architectures such assingle/multi-processor architectures and sequencers/parallelarchitectures, but also specialised circuits such as field programmablegate arrays FPGA, application specify circuits ASIC, signal processingdevices and other devices. References to computer program, instructions,code etc. should be understood to express software for a programmableprocessor firmware such as the programmable content of a hardware deviceas instructions for a processor or configured or configuration settingsfor a fixed function device, gate array, programmable logic device, etc.

As used in this application, the term “circuitry” refers to all of thefollowing: (a) hardware-only circuit implementations (such asimplementations in only analogue and/or digital circuitry) and (b) tocombinations of circuits and software (and/or firmware), such as (asapplicable): (i) to a combination of processor(s) or (ii) to portions ofprocessor(s)/software (including digital signal processor(s)), software,and memory(ies) that work together to cause an apparatus, such as aserver, to perform various functions) and (c) to circuits, such as amicroprocessor(s) or a portion of a microprocessor(s), that requiresoftware or firmware for operation, even if the software or firmware isnot physically present.

If desired, the different functions discussed herein may be performed ina different order and/or concurrently with each other. Furthermore, ifdesired, one or more of the above-described functions may be optional ormay be combined. Similarly, it will also be appreciated that the flowdiagram of FIG. 7 is an example only and that various operationsdepicted therein may be omitted, reordered and/or combined.

It will be appreciated that the above described example embodiments arepurely illustrative and are not limiting on the scope of the invention.Other variations and modifications will be apparent to persons skilledin the art upon reading the present specification.

Moreover, the disclosure of the present application should be understoodto include any novel features or any novel combination of featureseither explicitly or implicitly disclosed herein or any generalizationthereof and during the prosecution of the present application or of anyapplication derived therefrom, new claims may be formulated to cover anysuch features and/or combination of such features.

Although various aspects of the invention are set out in the independentclaims, other aspects of the invention comprise other combinations offeatures from the described embodiments and/or the dependent claims withthe features of the independent claims, and not solely the combinationsexplicitly set out in the claims.

It is also noted herein that while the above describes various examples,these descriptions should not be viewed in a limiting sense. Rather,there are several variations and modifications which may be made withoutdeparting from the scope of the present invention as defined in theappended claims.

The invention claimed is:
 1. An apparatus comprising: at least oneprocessor; and at least one memory including computer program code, theat least one memory and the computer program configured, with the atleast one processor, to cause the apparatus to perform at least:initialising trainable parameters of a transmission system, wherein thetransmission system comprises a transmitter, a channel and a receiver,wherein the transmitter includes a transmitter algorithm for convertingone or more inputs into one or more data symbols and a modulator forconverting said data symbols into transmit symbols in accordance with amodulation scheme, wherein the trainable parameters of the transmissionsystem include a probability function defining a probability thatindividual data symbols are output by the transmitter algorithm;generating training symbols on the basis of a distribution function,wherein the distribution function is differentiable; transmittingmodulated training symbols to the receiver over the channel in atraining mode; generating a loss function based on the generatedtraining symbols and the modulated training symbols as received at thereceiver of the transmission system; generating updated parameters ofthe transmission system in order to minimise the loss function, whereingenerating updated parameters of the transmission system compriseupdating the parameters of the transmitter algorithm; and repeating thegenerating the training symbols, generating the loss function andupdating parameters of the transmission system until a first conditionis reached.
 2. The apparatus as claimed in claim 1, wherein thedistribution function is an approximation of the probability function.3. The apparatus as claimed in claim 1, wherein the training symbols aregenerated by a Gumbel-softmax algorithm.
 4. The apparatus as claimed inclaim 1, wherein said transmit symbols correspond to fixed constellationpositions of a modulation scheme implemented by said modulator.
 5. Theapparatus as claimed in claim 1, wherein said transmit symbolscorrespond to variable constellation positions of a modulation schemeimplemented by said modulator, wherein the trainable parameters of thetransmission system comprise constellation point positions of saidmodulation scheme.
 6. The apparatus as claimed in claim 1, wherein thereceiver is configured to receive said transmit symbols as transmittedover said channel in an operational mode and to receive said modulatedtraining symbols as transmitted over the channel in a training mode. 7.The apparatus as claimed in claim 1, wherein generating updatedparameters of the transmission system comprises updating parameters of atrainable receiver algorithm.
 8. The apparatus as claimed in claim 1,wherein the at least one processor, with the at least one memory and thecomputer program code, is arranged to cause the apparatus to furtherperform: updating the trainable parameters of the transmission systemusing the generated updated parameters of the transmission system. 9.The apparatus as claimed in claim 8, wherein the trainable parameters ofthe transmitter system comprise one of more of: trainable parameters ofthe transmitter algorithm; trainable parameters of the modulator; andtrainable parameters of the receiver.
 10. The apparatus as claimed in aclaim 1, wherein the probability function and the distribution functionare at least partially dependent on channel information of thetransmission system.
 11. The apparatus as claimed in claim 1, whereinthe first condition comprises a defined performance level.
 12. Theapparatus as claimed in claim 1, wherein the first condition comprises adefined number of iterations.
 13. The apparatus as claimed in claim 1,wherein said transmitter algorithm is implemented as a look-up table inan operational mode.
 14. The apparatus as claimed in claim 1, whereinthe loss function is related to one or more of block error rate, biterror rate, mutual information and categorical cross-entropy.
 15. Theapparatus as claimed in a claim 1, wherein generating updated parametersof the transmission system optimises one or more of a batch size ofsymbol approximations used in the training mode and a learning rate. 16.The apparatus as claimed in claim 1, wherein generating updatedparameters of the transmission system updates said parameters usingstochastic gradient descent.
 17. The apparatus as claimed in claim 1,wherein the transmitter comprises a transmitter neural networkconfigured to implement said transmitter algorithm.
 18. A methodcomprising: initialising trainable parameters of a transmission system,wherein the transmission system comprises a transmitter, a channel and areceiver, wherein the transmitter includes a transmitter algorithm forconverting one or more inputs into one or more data symbols and amodulator for converting said data symbols into transmit symbols inaccordance with a modulation scheme, wherein the trainable parameters ofthe transmission system include a probability function defining aprobability that individual data symbols are output by the transmitteralgorithm; generating training symbols on the basis of a distributionfunction, wherein the distribution function is differentiable;transmitting modulated training symbols to the receiver over the channelin a training mode; generating a loss function based on the generatedtraining symbols and the modulated training symbols as received at thereceiver of the transmission system; generating updated parameters ofthe transmission system in order to minimise the loss function, whereingenerating updated parameters of the transmission system compriseupdating the parameters of the transmitter algorithm; and repeating thegenerating the training symbols, generating the loss function andupdating parameters of the transmission system until a first conditionis reached.
 19. A method as claimed in claim 18, wherein: said transmitsymbols correspond to fixed constellation positions of a modulationscheme implemented by said modulator; or said transmit symbolscorrespond to variable constellation positions of a modulation schemeimplemented by said modulator, wherein the trainable parameters of thetransmission system comprise constellation point positions of saidmodulation scheme.
 20. A non-transitory computer readable mediumcomprising instructions that, when executed by an apparatus, cause theapparatus to at least: initialise trainable parameters of a transmissionsystem, wherein the transmission system comprises a transmitter, achannel and a receiver, wherein the transmitter includes a transmitteralgorithm for converting one or more inputs into one or more datasymbols and a modulator for converting said data symbols into transmitsymbols in accordance with a modulation scheme, wherein the trainableparameters of the transmission system include a probability functiondefining a probability that individual data symbols are output by thetransmitter algorithm; generate training symbols on the basis of adistribution function, wherein the distribution function isdifferentiable; transmit modulated symbol to the receiver over thechannel in a training mode; generate a loss function based on thegenerated symbol and the modulated symbol as received at the receiver ofthe transmission system; generate updated parameters of the transmissionsystem in order to minimise the loss function, wherein generatingupdated parameters of the transmission system comprise updating theparameters of the transmitter algorithm; and repeat the generating thesymbol approximations, generating the loss function and updatingparameters of the transmission system until a first condition isreached.